Phone: 212-766-2120 cell: 917-494-1606. Agenda Creating our Learning Community & Norms Setting...

phone: 212-766-2120

cell: 917-494-1606

AgendaCreating our Learning Community & NormsSetting Personal Goals and Identifying

ThemesVisualization through Quick Images’Math Routines that Develop ThinkingExploring Addition and Mental MathLunchGames

Eight CCSS Math PracticesMake sense of problems and persevere in

solving them. Reason abstractly and quantitatively.Construct viable arguments and critique the

reasoning of others. Model with mathematics.Use appropriate tools strategically.Attend to precision.Look for and make use of structure.Look for and express regularity in repeated

reasoning.

Construct Viable Arguments and Critique the Reasoning of OthersUnderstand and use stated assumptions,

definitions and previously established resultsMake and explore conjectures using a logical

progression of statementsAnalyze arguments using cases,

counterexamples, data, contextsCompare effectiveness of two plausible

argumentsDistinguish correct from flawed reasoning

Model with MathematicsApply mathematics to solve everyday, society,

and workplace problemsMake assumptions and approximations to

simplify a complicated situation and revise as needed

Map quantitative relationships in situations using diagrams, charts, flow-charts, tables, etc.

Analyze quantitative relationships to draw conclusions

Interpret results in relation to the context determining whether answer makes sense or model needs revision

Developing Addition Strategies

Today’s Session Will Explore …

What does it mean to have computational fluency?

How do we develop computational fluency in our students?


What is in our addition toolbox?


39 + 68

Share your strategies in a small group.

Are your strategies the same? Different?


39 + 68

What mathematical ideas make these strategies work?


Did you use the standard algorithm?

39 + 68


1

39 + 68

107

What mathematical ideas make the standard addition algorithm work?



What mathematical ideas make the standard addition algorithm work? Knowing that place determines value

Equivalence Associativity Commutativity Unitizing

Strategy Mathematical Ideas What it looks like …

Standard Addition

Algorithm

Place determines value 39 + 68 = (30 + 9) +( 60 + 8)

Associative property39 + 68 = (30 + 9) +( 60 + 8) (9 + 8) +( 30 + 60)

Commutative property 39 + 68 = 68 + 39

UnitizingSaying, “3” + “6” in the algorithm MEANS3 (tens) + 6 (tens)

Equivalence 39 = 30 + 9 = 20 + 19 = 10 + 29


1 39

+ 68 107

(8+9) + (60 + 30) =17 + 60 + 30 = 10 + 7 + 60 + 30 = 90 + 10 + 7 = 107


Other possible strategies for adding 39 + 68?

Partial sums (splitting)

Creating an equivalent problem (compensation)

Keeping one number whole and adding in parts

Using the ten structure of the number system to

Move to landmarks

Take landmark jumps



Partial sums (splitting)

(30 + 9) + (60 + 8) = (30 + 60) + (9 + 8) =90 + 17 = (90 + 10) + 7 =100 + 7 = 107



Compensation (creating an equivalent problem)

39 + 68 = 39 + (67 +1) = 39 + (1 + 67) = (39 + 1) + 67 = 40 + 67 = 107

39 + 68 = (37 + 2) + 68 = 37 + (2 + 68) = 37 + 70 = 37 + 70 = 107



Keeping one number whole and adding in parts:

39 + 68 =

68 + (30 + 9) = (68 + 30) + 998 + 9 = 107



Making or using landmark jumps:

39 + 68 = 68 + 3968 + (20 + 10 + 9) =

(68 + 20) + 10 + 9 = (88 + 10) + 9 = 98 + (10 – 1) = 107



Moving to the nearest landmark:

39 + 68 = 39 + 1 + 67 =40 + 67 = 40 + (60 + 7)(40 + 60) + 7 = 100 + 7 = 107

Addition Strategies What they look like …

partial sums (splitting)

39 + 68 = (30 + 9) +( 60 + 8) = (30 + 60) +( 9 + 8)

compensation 39 + 68 = 39 + (67 +1) = (39 + 1) + 67 = 40 + 67

keeping one number whole

39 + 68 = 68 + (30 + 9) = 98 + 9 = 107

making landmarks jumps

39 + 68 = 68 + 39 = (68 + 20) + 10 + 9 = 88 + 10 + 9 = 98 + (10 – 1) = 107

moving to the nearest landmarks

39 + 68 = 39 + 1 + 67 = (40 + 60) + 7 = 107


It’s important to notice that many of the big ideas underlying these addition strategies are the same:

Big ideas: 1.Equivalence 39 + 68 = 40 + 672.Commutative property: a + (b + c) = a + (c + b)3.Associative property of addition: a + (c + b) = (a

+ c) + b4.Place determines value (the “3” in 39 is 30 or 3

tens)


All of these strategies can be used algorithmically.

The key is not just to have alternative mental-math strategies, but to know when to use them.


Why?


One of the hallmarks of number sense is flexible strategy use.

What does this mean?

In computation, it means looking to the numbers to pick the most efficient strategy.


HOW TO WE DEVELOP FLEXIBLE

STRATEGY USE IN OUR STUDENTS?


One way to help students develop important number relationships is through computational mini-lessons.


Guided Mini-lessons

“Strings”


Strings are a series of interconnected bare

number problems which teachers design and

modify ad hoc in order to help students invent

and/or use efficient mental-math computation

strategies.


What mental–math strategy might a teacher using this mini-lesson be developing?

43 + 20

62 + 30

62 + 39

54 + 48



To successfully use mental-math mini-lessons, one must consider

• The role of the student• The role of the teacher• The role (and power) of mathematical models

The Role of the Student

Students are expected to1. Find their own solutions to the problem2. Share their thinking publicly3. Listen to and make sense of the strategies

of others4. Find and pose questions when they don’t

understand or they need clarification5. Try on new strategies6. Practice until strategies become automatic

The Role of Student Discourse

Why is talk so important to the development of computational strategies?1. Each time a strategy is discussed, students gain

additional insights through other children’s explanations.

2. Over time, both through listening and questioning, students eventually make sense of the strategy and begin to feel comfortable with the strategy.

3. The students then may attempt to use the strategy in some situations.

4. Over time and with use, the strategy then becomes integrated into students’ mental-math repertoires and is used regularly when needed.

The Role of the Teacher

Encouraging students to make sense of situations;Providing time for students to question each other’s

thinking and strategies;Connecting different strategies to help students

understand each other’s thinking;Highlighting efficient strategies;Using questions such as:

How did you get your answer?Can you explain it another way?Did anyone do it the same way? Can you put in your own words _______’s thinking?

The Role of Mathematical Models

Models become important tools to represent student thinking.

Models become important tools to connect and juxtapose student strategies.

Models become important tools for students to think with.


Addition Strings

What addition strategies are the strings on the worksheet designed to develop?

Developing Addition StrategiesWhat strategy is the string trying to develop?

54 + 2055 + 1942 + 4044 + 3866 +3069 +2789 +73


Choose one of the strings from the handout with a partner to analyze for:

(1) potential student strategies and struggles; (2) how to model or represent their thinking; (3) what questions to pose to students given their respective strategies or struggles.

GamesWhy use games to teach math?What do teachers need to do to ensure

students are actually attending to the math in the game?

How can games be used to differentiate?What kind of record keeping system would

help us keep track of student learning when playing games?

How might we organize games for the greatest student autonomy and make sure they work with “just right” games as determined by informal assessment?

Games

GamesChoose a game and play it with a partner or

in a small group.Use the games analysis sheet to name the

mathematics in the game, compare with games you already use,consider various ways to extend the game or make it less challenging.

Use the Games Continuum sheet to help you place the game on the continuum and begin building your games library.

Games ContinuumCounting Comparing Counting

On/Conservation

Games That Use 5/10 Structure

Part Whole Relations and Equivalence

Roll and Record

Build It

Compare/Top It/War

Racing Bears

Ten Frame Match

Counters in a Cup

Games ContinuumAddition or Fact Fluency

Subraction Games

Money Games

Combinations of Tens Games

Number Line Games

Addition Games

Place Value Games

Roll and Record

Build It

Compare/Top It/War

Racing Bears

Ten Frame Match

Counters in a Cup

ReflectionsWhat are you questions and takeaways from

today’s session? Thank you!

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Phone: 212-766-2120 cell: 917-494-1606. Agenda Creating our Learning Community & Norms Setting...

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